2 Nonstrict solutions of the bisymmetry equation

On Nonstrict Means

J´anos C. Fodor Department of Mathematics

University of Agricultural Sciences, G¨od¨oll˝o, P´ater K. u. 1, H-2103 G¨od¨oll˝o, Hungary

Jean-Luc Marichal

Ecole d’Administration des Affaires, University of Li`ege 7, boulevard du Rectorat - B31, B-4000 Li`ege 1, Belgium

∗

Abstract The general form of continuous, sym- metric, increasing, idempotent solutions of the bisymmetry equation is established and the fam- ily of sequences of functions which are continuous, symmetric, increasing, idempotent, decomposable is described.

Keywords: nonstrict mean values, bisymmetry equation, decomposability property, ordinal sums.

1 Introduction

Kolmogoroff [6] and Nagumo [8] established a fun- damental result about mean values. In their defi- nition a mean value is a sequence (M^(m))m∈IN0 of functions M^(m) : [a, b]^m → [a, b] (where [a, b] is a closed real interval) which are continuous, sym- metric, strictly increasing in each argument, and idempotent (that is M^(m)(x, . . . , x) = x for all x ∈ [a, b]). These functions are also linked by a pseudo-associativity called the decomposability property by several authors (see e.g. [3, Chapter 5]):

M^(m)(x1, . . . , xk, xk+1, . . . , xm)

= M^(m)(Mk, . . . , Mk, xk+1, . . . , xm) for allm∈IN0,k∈ {1, . . . , m},x1, . . . , xm∈[a, b], withMk=M^(k)(x1, . . . , xk).

The corresponding result of Kolmogoroff and Nagumo states that these conditions are neces- sary and sufficient for the existence of a continuous strictly monotonic real functionf such that

M^(m)(x1, . . . , xm) =f⁻¹

"

1 m

Xm

i=1

f(xi)

#

for all m ∈ IN0. Such an expression is called the generalized mean.

On the other hand, Acz´el [1] (see also [2]) proved that a function M of two variables defined on [a, b] is continuous, symmetric, strictly increasing in each argument, idempotent and fulfils thebisym- metry equation

M[M(x₁₁, x₁₂), M(x₂₁, x₂₂)]

=M[M(x11, x21), M(x12, x22)] (1) if and only if

M(x, y) =f⁻¹

·f(x) +f(y) 2

¸

, x, y∈[a, b]

with some continuous strictly monotonic function f.

Note that Horv´ath [5] investigated the connec- tion between the two concepts of bisymmetry and decomposability.

The aim of this paper is to study nonstrict means in an elementary way. That is, we in- vestigate means satisfying either the conditions of Acz´el’s theorem or the conditions of Kolmogo- roff and Nagumo’s theorem above, except strict monotonicity. We describe the family of contin- uous, symmetric, increasing, idempotent, bisym- metric functions (Section 2) and also the family of sequences of continuous, symmetric, increasing, idempotent and decomposable functions (Section 3). The general families obtained are very similar to those ones well-known as ordinal sums in the theory of the associativity functional equation (see e.g. [7]). For space limitation, no proof of the re- sults will be given.

1

2 Nonstrict solutions of the bisymmetry equation

The bisymmetry equation (1), which can be consid- ered also as a generalization of simultaneous com- mutativity and associativity, has been investigated by several authors. For a list of references see [2].

A functionM : [a, b]²→[a, b] is called

• symmetric ifM(x, y) =M(y, x) for allx, y ∈ [a, b];

• increasing if x≤ x⁰, y ≤y⁰ imply M(x, y) ≤ M(x⁰, y⁰);

• strictly increasingifx < x⁰ impliesM(x, y)<

M(x⁰, y) and the same fory < y⁰;

• idempotentifM(x, x) =xfor allx∈[a, b].

As said above, our goal is to describe the general form of continuous, symmetric, increasing, idem- potent solutions of the bisymmetry equation (1).

In their structures, these solutions are very similar to ordinal sums which are well-known in the theory of semigroups, see e.g. [7].

Acz´el [1] proved the following result.

Theorem 1 M : [a, b]² → [a, b] is a continuous, symmetric, strictly increasing, idempotent, bisym- metric function if and only if

M(x, y) =f⁻¹

·f(x) +f(y) 2

¸

(generalized mean) where f is any continuous strictly monotonic function on[a, b].

We also know that this result still holds for in- tervals of the form (a, b), [a, b), (a, b] or even for any unbounded interval of the real line (see [2], pp 250–251, 280).

Now define B_a,b,θ as the set of functions M : [a, b]² → [a, b] which are continuous, symmetric, increasing, idempotent, bisymmetric and such that M(a, b) = θ, θ being a given number in [a, b]. A general element of a classBa,b,θis usually denoted byMa,b,θin the sequel. Then we have the following result.

Theorem 2 M : [a, b]² → [a, b] is a continu- ous, symmetric, increasing, idempotent, bisymmet- ric function if and only if there exists two numbers αandβ fulfillinga≤α≤β ≤bsuch that

i) M(x, y) =Ma,α,α(x, y) ifx, y ∈[a, α];

ii) M(x, y) =Mβ,b,β(x, y) ifx, y ∈[β, b];

iii) M(x, y) = f⁻¹

·f[median(α, x, β)] +f[median(α, y, β)]

2

¸

otherwise,

with some Ma,α,α ∈ Ba,α,α, Mβ,b,β ∈ Bβ,b,β, and f is any continuous strictly monotonic function on [α, β].

Now, describe the two families Ba,b,a andBa,b,b. Theorem 3 We have M ∈ Ba,b,a if and only if

• either

M(x, y) = min(x, y),

• or

M(x, y) =g⁻¹p

g(x)g(y),

where g is any continuous strictly increasing function on[a, b], withg(a) = 0,

• or there exists a countable index set K and a family of disjoint subintervals {(ak, bk) :k ∈ K} of[a, b] such that

M(x, y) =











 g⁻¹_k p

gk[min(x, bk)]gk[min(y, bk)]

if there exists k∈K such that min(x, y)∈(ak, bk),

min(x, y) otherwise,

wheregk is any continuous strictly increasing function on[ak, bk], with gk(ak) = 0.

Theorem 4 We have M ∈ Ba,b,b if and only if

• either

M(x, y) = max(x, y),

• or

M(x, y) =g⁻¹p

g(x)g(y),

where g is any continuous strictly decreasing function on[a, b], withg(b) = 0,

• or there exists a countable index set K and a family of disjoint subintervals {(ak, bk) :k ∈ K} of[a, b] such that

M(x, y) =











 g⁻¹_k p

gk[max(ak, x)]gk[max(ak, y)]

if there exists k∈K such that max(x, y)∈(ak, bk),

max(x, y)otherwise,

wheregk is any continuous strictly decreasing function on[ak, bk], with gk(bk) = 0.

2

3 Extended

Kolmogoroff-means

We show now that the results obtained in the pre- vious section can be extend to the mean values by replacing bisymmetry by decomposability. Ac- cording to Fodor and Roubens [3], we can see any mean valueM as an aggregation operator:

M : Λ = [∞

m=1

[a, b]^m → [a, b]

(x1, . . . , xm) → x=M^(m)(x1, . . . , xm)¹ Such an operatorM is said to be

• continuous if for all m ∈IN0, M(x1, . . . , xm) is a continuous function on [a, b]^m;

• symmetric if for all m ∈ IN0, M(x1, . . . , xm) is a symmetric function on [a, b]^m:

M(x1, . . . , xm) =M(x_σ(1), . . . , x_σ(m)) whereσis a permutation of{1, . . . , m};

• increasingif for allm∈IN₀,M(x₁, . . . , x_m) is increasing in each argument:

xi< x⁰_i ⇒ M(x1, . . . , xi, . . . , xm)

≤M(x1, . . . , x⁰_i, . . . , xm), fori= 1, . . . , m;

• strictly increasing if for all m ∈ IN0, M(x1, . . . , xm) is strictly increasing in each ar- gument:

xi< x⁰_i ⇒ M(x1, . . . , xi, . . . , xm)

< M(x1, . . . , x⁰_i, . . . , xm), fori= 1, . . . , m;

• idempotentif for allm∈IN0,M has to satisfy M^(m)(x, . . . , x) =x, ∀x∈[a, b];

• decomposable if for all m ∈ IN0 and all k ∈ {1, . . . , m},M has to satisfy

M^(m)(x1, . . . , xk, xk+1, . . . , xm)

= M^(m)(Mk, . . . , Mk, xk+1, . . . , xm) whereMk=M^(k)(x1, . . . , xk).

Kolmogoroff [6] established the following result.

Theorem 5 An aggregation operator M, defined onΛ, is continuous, symmetric, strictly increasing, idempotent and decomposable if and only if for all m∈IN0,

M^(m)(x1, . . . , xm) =f⁻¹

"

1 m

X

i

f(xi)

#

(generalized mean) where f is any continuous strictly monotonic function on[a, b].

Theorem 5 still holds for sets Λ of the form S_∞

m=1(a, b)^m, S_∞

m=1[a, b)^m, S_∞

m=1(a, b]^m, even if a=−∞and/orb= +∞.

DefineDa,b,θas the set of aggregation operators M : Λ = S_∞

m=1[a, b]^m → [a, b] which are contin- uous, symmetric, increasing, idempotent, decom- posable and such thatM(a, b) =θ,θbeing a given number in [a, b]. Then we have the following result:

Theorem 6 An aggregation operator M, defined on Λ, is continuous, symmetric, increasing, idem- potent and decomposable if and only if there exists two numbers α and β fulfilling a ≤ α ≤ β ≤ b, such that, for all m∈IN0,

i) M(x1, . . . , xm) =Ma,α,α(x1, . . . , xm) if maxixi∈[a, α];

ii) M(x1, . . . , xm) =Mβ,b,β(x1, . . . , xm) if minixi ∈[β, b];

iii) M(x1, . . . , xm) = f⁻¹

"

1 m

X

i

f[median(α, xi, β)]

#

otherwise,

whereMa,α,α∈ Da,α,α,Mβ,b,β ∈ Dβ,b,β and where f is any continuous strictly monotonic function on [α, β].

Now, describe the two families Da,b,a andDa,b,b. Theorem 7 We have M ∈ Da,b,a if and only if for all m∈IN0,

• either

M(x1, . . . , xm) = min

i xi,

• or

M(x1, . . . , xm) =g^−1msY

i

g(xi),

where g is any continuous strictly increasing function on[a, b], withg(a) = 0,

3

• or there exists a countable index set K and a family of disjoint subintervals {(ak, bk) : k ∈ K}of [a, b] such that

M(x1, . . . , xm) =

















g⁻¹_k ^mpQ

igk[min(xi, bk)]

if there exists k∈K such that

minixi∈(ak, bk), minixi otherwise, wheregk is any continuous strictly increasing function on[a_k, b_k], withg_k(a_k) = 0.

Theorem 8 We have M ∈ Da,b,b if and only if for allm∈IN0,

• either

M(x1, . . . , xm) = max

i xi,

• or

M(x1, . . . , xm) =g^−1msY

i

g(xi),

where g is any continuous strictly decreasing function on[a, b], with g(b) = 0,

• or there exists a countable index set K and a family of disjoint subintervals {(ak, bk) : k ∈ K}of [a, b] such that

M(x1, . . . , xm) =

















g⁻¹_k ^mpQ

igk[max(ak, xi)]

if there exists k∈K such that

maxixi∈(ak, bk), maxixi otherwise, wheregk is any continuous strictly decreasing function on[ak, bk], withgk(bk) = 0.

References

[1] J. Acz´el (1948), On mean values, Bulletin of the American Math. Society,54: 392-400.

[2] J. Acz´el (1966), Lectures on Functional Equa- tions and Applications,(Academic Press, New York).

[3] J. Fodor and M. Roubens (1994), Fuzzy Pref- erence Modelling and Multicriteria Decision Support,Kluwer, Dordrecht.

[4] L. Fuchs (1950), On mean systems, Acta Math. Acad. Sci. Hung.1: 303-320.

[5] J. Horv´ath (1947), Sur le rapport entre les syst`emes de postulats caract´erisant les valeurs moyennes quasi arithm´etiques sym´etriques,C.

R. Acad. Sci. Paris,225: 1256-1257.

[6] A.N. Kolmogoroff (1930), Sur la notion de la moyenne, Accad. Naz. Lincei Mem. Cl. Sci.

Fis. Mat. Natur. Sez.,12: 388-391.

[7] C.H. Ling (1965), Representation of associa- tive functions,Publ. Math. Debrecen,12: 189- 212.

[8] M. Nagumo (1930), ¨Uber eine Klasse der Mit- telwerte,Japanese Journal of Mathematics,6:

71-79.

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